176-midterm-solutions-03-03-11 - Physics 176 Midterm Exam...

Physics 176: Midterm Exam SolutionsMarch, 2011Most of the following answers are more detailed than what was necessary to get full credit. I hope the detailswill help you improve your problem solving skills and understand the physics better.Problems That Require Writing1. Black holes are simple pure objects in that, no matter what kind of matter collapsed to form the blackhole, only three numbers are needed to define its macrostate: the black hole’s massM, its electricalchargeQ, and its angular momentumL. For an electrically-neutral non-rotating black hole (Q= 0,L=0), the black hole’s entropySdepends only onMand is given byS=8π2kGhcM2(1)wherekis Boltzmann’s constant,Gis the gravitational constant,his Planck’s constant, andcis thespeed of light.(a)(5 points)Given that the energy of a black hole is its relativistic rest massU=Mc2, derive aformula for the temperatureTof a black hole in terms of its massM. Does decreasing the massof a black hole make it hotter or colder?Answer:The temperature can be deduced from the dependence of the entropyS(U) on thesystem’s energyUusing the standard formula. We have:1T=∂S∂U(2)=∂S∂M×dMdU(3)=8π2kGhc·2M×1c2(4)=16π2kGhc3M,(5)so the answer isT=hc316π2GkM∝M-1.(6)Note how the entropy and temperature of a black hole involve fundamental constants from fourdifferent areas of physics: thermal physics (k), gravity (G), quantum mechanics (h), and specialrelativity (c). This makes black holes intriguing indeed.SinceT∝M-1and mass is always a positive quantity (any material object resists acceleration),the temperature must also be positive for a black hole. Black holes are exotic objects but theycan’t have a negative temperature like a paramagnet.BecauseT∝M-1, the smaller the mass of a black hole, the hotter it is.A sufficiently smallblack hole (about the mass of the Earth’s moon or less, 1022kg) will become hotter than 3 Kthe temperature of the cosmic blackbody radiation.We would then expect energy to transferfrom the black hole to the surrounding photon gas if the black hole could transfer energy. Thishas been predicted to be possible by the physicist Stephen Hawking, who heuristically appliedrelativistic quantum mechanics to Einstein’s classical gravity theory to predict that all black holesemit light radiation from their event horizon (called “Hawking radiation”) and so can evaporateaway into light if hotter than their surroundings. Further, the smaller the mass of a black hole,the faster it loses mass by Hawking radiation. The tiny small mass black holes predicted to occur1

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in the Large Hadron Collider (based on the assumption that there are more than three spatialdimensions) are super hot and disappear within 10-10s of their appearance, in a burst of gammarays which would be the signature for their appearance. You can learn more about Eq. (6), calledthe Hawking radiation temperature, from the Wikipedia article with title “Hawking Radiation”.

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